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Proving Popescu approximation reduces to algebras over a field

Lemma 16.8.4. If for every Situation 16.8.1 where $R$ is a field PT holds, then PT holds in general.

Proof. Assume PT holds for any Situation 16.8.1 where $R$ is a field. Let $R \to \Lambda $ be as in Situation 16.8.1 arbitrary. Note that $R/I \to \Lambda /I\Lambda $ is another regular ring map of Noetherian rings, see More on Algebra, Lemma 15.41.3. Consider the set of ideals

\[ \mathcal{I} = \{ I \subset R \mid R/I \to \Lambda /I\Lambda \text{ does not have PT}\} \]

We have to show that $\mathcal{I}$ is empty. If this set is nonempty, then it contains a maximal element because $R$ is Noetherian. Replacing $R$ by $R/I$ and $\Lambda $ by $\Lambda /I$ we obtain a situation where PT holds for $R/I \to \Lambda /I\Lambda $ for any nonzero ideal of $R$. In particular, we see by applying Proposition 16.5.3 that $R$ is a reduced ring.

Let $A \to \Lambda $ be an $R$-algebra homomorphism with $A$ of finite presentation. We have to find a factorization $A \to B \to \Lambda $ with $B$ smooth over $R$, see Algebra, Lemma 10.127.4.

Let $S \subset R$ be the set of nonzerodivisors and consider the total ring of fractions $Q = S^{-1}R$ of $R$. We know that $Q = K_1 \times \ldots \times K_ n$ is a product of fields, see Algebra, Lemmas 10.25.4 and 10.31.6. By Lemma 16.8.2 and our assumption PT holds for the ring map $S^{-1}R \to S^{-1}\Lambda $. Hence we can find a factorization $S^{-1}A \to B' \to S^{-1}\Lambda $ with $B'$ smooth over $S^{-1}R$.

We apply Lemma 16.8.3 and find a factorization $A \to B \to \Lambda $ such that some $\pi \in S$ is elementary standard in $B$ over $R$. After replacing $A$ by $B$ we may assume that $\pi $ is elementary standard, hence strictly standard in $A$. We know that $R/\pi ^8R \to \Lambda /\pi ^8\Lambda $ satisfies PT. Hence we can find a factorization $R/\pi ^8 R \to A/\pi ^8A \to \bar C \to \Lambda /\pi ^8\Lambda $ with $R/\pi ^8 R \to \bar C$ smooth. By Lemma 16.6.1 we can find an $R$-algebra map $D \to \Lambda $ with $D$ smooth over $R$ and a factorization $R/\pi ^4 R \to A/\pi ^4A \to D/\pi ^4D \to \Lambda /\pi ^4\Lambda $. By Lemma 16.7.2 we can find $A \to B \to \Lambda $ with $B$ smooth over $R$ which finishes the proof. $\square$

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